Complex Analysis Script

Note.
The Complex Analysis Script was prepared by Prof. Schep for the analysis qual syllabus effective 1995.
Some parts of this script will be omitted for the new analysis syllabus effective 2020.

Lecture Notes

Ch. 1. Holomorphic (i.e., Analytic) Functions 1.1 Definitions and elementary properties (pg. 1-5) 1.2 Elementary transcendental functions (pg. 6-9) 1.3 Differentiable functions (pg. 10-21) 1.4 Power Series (pg. 22-29) Ch. 2. Integration over contours 2.1 Curves and Contours (pg. 30-33) 2.2 Contour/Path Integrals Part 1: pg. 33-38 2.2 Contour/Path Integrals Part 2: pg. 39-48 2.2 Contour/Path Integrals Part 3: pg. 49-56 2.2 Contour/Path Integrals Part 4: pg. 56-62 Ch. 3. Zeros and singularities of holomorphic function (to be continued) 3.1 Zeros of holomorphic functions Part 1: pg. 1-6 3.2 Singularities of holomorphic functions 3.3 The Residue Theorem and Applications

Further Reference for Complex Analysis  (not required)

As listed on the Analysis Qual Syllabus (listed alphabetically by author): Ahlfors L., Complex Analysis (3rd edition), McGraw-Hill, New York, 1978. Conway J., Functions of One Complex Variable, Springer-Verlag, New York, 1978. Rudin W., Real and Complex Analysis, McGraw-Hill, New York, 1987. Saks and Zygmund, Analytic Functions, Elsevier, New York, 1971. Others: Complex Variables and Applications by Churchill and Brown (any edition). (A good undergraduate textbook.) Schaum's Outline on Complex Variables (any edition). (has many problems with solutions) Complex Variables by Robert B. Ash and W. Phil Novinger. 2nd Edition. Dover Publications. 2007. This book is available, as several PDF files, from Prof. Ash's homepage. Here is the book as one PDF file with hyperlinks. (Hyperlinks were added by Prof. Girardi. Please email her needed corrections you notice.) Visual Complex Analysis by Tristan Needham. (gives great geometrical intuition, lacks rigor).